Beg surgeon to work around top of cyst (8)
I believe the answer is:
scrounge
'beg' is the definition.
(scrounging is a kind of begging)
'surgeon to work around top of cyst' is the wordplay.
'to work' is an anagram indicator.
'around' indicates putting letters inside.
'top of' says to take the initial letters.
The initial letter of 'cyst' is 'c'.
'surgeon' anagrammed gives 'srounge'.
'srounge' going around 'c' is 'SCROUNGE'.
(Other definitions for scrounge that I've seen before include "Cadge, sponge" , "Freeload" , "Acquire by begging" , "Cur's gone (anag) - beg" , "bum" .)